Add Covariance documentation to html docs.
Change-Id: I11ddc9f7069964596760c6ea4d85c44312c0a67a
diff --git a/docs/source/bibliography.rst b/docs/source/bibliography.rst
index 188ed77..30639d1 100644
--- a/docs/source/bibliography.rst
+++ b/docs/source/bibliography.rst
@@ -39,6 +39,11 @@
.. [HartleyZisserman] R.I. Hartley & A. Zisserman, **Multiview
Geometry in Computer Vision**, Cambridge University Press, 2004.
+.. [KanataniMorris] K. Kanatani and D. D. Morris, **Gauges and gauge
+ transformations for uncertainty description of geometric structure
+ with indeterminacy**, *IEEE Transactions on Information Theory*
+ 47(5):2017-2028, 2001.
+
.. [KushalAgarwal] A. Kushal and S. Agarwal, **Visibility based
preconditioning for bundle adjustment**, *In Proceedings of the
IEEE Conference on Computer Vision and Pattern Recognition*, 2012.
diff --git a/docs/source/solving.rst b/docs/source/solving.rst
index e26f87b..80aee86 100644
--- a/docs/source/solving.rst
+++ b/docs/source/solving.rst
@@ -1678,8 +1678,350 @@
};
+Covariance Estimation
+=====================
-:class:`GradientChecker`
-------------------------
+Background
+----------
-.. class:: GradientChecker
+One way to assess the quality of the solution returned by a
+non-linear least squares solve is to analyze the covariance of the
+solution.
+
+Let us consider the non-linear regression problem
+
+.. math:: y = f(x) + N(0, I)
+
+i.e., the observation :math:`y` is a random non-linear function of the
+independent variable :math:`x` with mean :math:`f(x)` and identity
+covariance. Then the maximum likelihood estimate of :math:`x` given
+observations :math:`y` is the solution to the non-linear least squares
+problem:
+
+.. math:: x^* = \arg \min_x \|f(x)\|^2
+
+And the covariance of :math:`x^*` is given by
+
+.. math:: C(x^*) = \left(J'(x^*)J(x^*)\right)^{-1}
+
+Here :math:`J(x^*)` is the Jacobian of :math:`f` at :math:`x^*`. The
+above formula assumes that :math:`J(x^*)` has full column rank.
+
+If :math:`J(x^*)` is rank deficient, then the covariance matrix :math:`C(x^*)`
+is also rank deficient and is given by the Moore-Penrose pseudo inverse.
+
+.. math:: C(x^*) = \left(J'(x^*)J(x^*)\right)^{\dagger}
+
+Note that in the above, we assumed that the covariance matrix for
+:math:`y` was identity. This is an important assumption. If this is
+not the case and we have
+
+.. math:: y = f(x) + N(0, S)
+
+Where :math:`S` is a positive semi-definite matrix denoting the
+covariance of :math:`y`, then the maximum likelihood problem to be
+solved is
+
+.. math:: x^* = \arg \min_x f'(x) S^{-1} f(x)
+
+and the corresponding covariance estimate of :math:`x^*` is given by
+
+.. math:: C(x^*) = \left(J'(x^*) S^{-1} J(x^*)\right)^{-1}
+
+So, if it is the case that the observations being fitted to have a
+covariance matrix not equal to identity, then it is the user's
+responsibility that the corresponding cost functions are correctly
+scaled, e.g. in the above case the cost function for this problem
+should evaluate :math:`S^{-1/2} f(x)` instead of just :math:`f(x)`,
+where :math:`S^{-1/2}` is the inverse square root of the covariance
+matrix :math:`S`.
+
+Gauge Invariance
+----------------
+
+In structure from motion (3D reconstruction) problems, the
+reconstruction is ambiguous upto a similarity transform. This is
+known as a *Gauge Ambiguity*. Handling Gauges correctly requires the
+use of SVD or custom inversion algorithms. For small problems the
+user can use the dense algorithm. For more details see the work of
+Kanatani & Morris [KanataniMorris]_.
+
+
+:class:`Covariance`
+-------------------
+
+:class:`Covariance` allows the user to evaluate the covariance for a
+non-linear least squares problem and provides random access to its
+blocks. The computation assumes that the cost functions compute
+residuals such that their covariance is identity.
+
+Since the computation of the covariance matrix requires computing the
+inverse of a potentially large matrix, this can involve a rather large
+amount of time and memory. However, it is usually the case that the
+user is only interested in a small part of the covariance
+matrix. Quite often just the block diagonal. :class:`Covariance`
+allows the user to specify the parts of the covariance matrix that she
+is interested in and then uses this information to only compute and
+store those parts of the covariance matrix.
+
+Rank of the Jacobian
+--------------------
+
+As we noted above, if the Jacobian is rank deficient, then the inverse
+of :math:`J'J` is not defined and instead a pseudo inverse needs to be
+computed.
+
+The rank deficiency in :math:`J` can be *structural* -- columns
+which are always known to be zero or *numerical* -- depending on the
+exact values in the Jacobian.
+
+Structural rank deficiency occurs when the problem contains parameter
+blocks that are constant. This class correctly handles structural rank
+deficiency like that.
+
+Numerical rank deficiency, where the rank of the matrix cannot be
+predicted by its sparsity structure and requires looking at its
+numerical values is more complicated. Here again there are two
+cases.
+
+ a. The rank deficiency arises from overparameterization. e.g., a
+ four dimensional quaternion used to parameterize :math:`SO(3)`,
+ which is a three dimensional manifold. In cases like this, the
+ user should use an appropriate
+ :class:`LocalParameterization`. Not only will this lead to better
+ numerical behaviour of the Solver, it will also expose the rank
+ deficiency to the :class:`Covariance` object so that it can
+ handle it correctly.
+
+ b. More general numerical rank deficiency in the Jacobian requires
+ the computation of the so called Singular Value Decomposition
+ (SVD) of :math:`J'J`. We do not know how to do this for large
+ sparse matrices efficiently. For small and moderate sized
+ problems this is done using dense linear algebra.
+
+
+:class:`Covariance::Options`
+
+.. class:: Covariance::Options
+
+.. member:: int Covariance::Options::num_threads
+
+ Default: ``1``
+
+ Number of threads to be used for evaluating the Jacobian and
+ estimation of covariance.
+
+.. member:: bool Covariance::Options::use_dense_linear_algebra
+
+ Default: ``false``
+
+ When ``true``, ``Eigen``'s ``JacobiSVD`` algorithm is used to
+ perform the computations. It is an accurate but slow method and
+ should only be used for small to moderate sized problems.
+
+ When ``false``, ``SuiteSparse/CHOLMOD`` is used to perform the
+ computation. Recent versions of ``SuiteSparse`` (>= 4.2.0) provide
+ a much more efficient method for solving for rows of the covariance
+ matrix. Therefore, if you are doing large scale covariance
+ estimation, we strongly recommend using a recent version of
+ ``SuiteSparse``.
+
+ This setting also has an effect on how the following two options
+ are interpreted.
+
+.. member:: int Covariance::Options::min_reciprocal_condition_number
+
+ Default: :math:`10^{-14}`
+
+ If the Jacobian matrix is near singular, then inverting :math:`J'J`
+ will result in unreliable results, e.g, if
+
+ .. math::
+
+ J = \begin{bmatrix}
+ 1.0& 1.0 \\
+ 1.0& 1.0000001
+ \end{bmatrix}
+
+ which is essentially a rank deficient matrix, we have
+
+ .. math::
+
+ (J'J)^{-1} = \begin{bmatrix}
+ 2.0471e+14& -2.0471e+14 \\
+ -2.0471e+14 2.0471e+14
+ \end{bmatrix}
+
+ This is not a useful result.
+
+ The reciprocal condition number of a matrix is a measure of
+ ill-conditioning or how close the matrix is to being singular/rank
+ deficient. It is defined as the ratio of the smallest eigenvalue of
+ the matrix to the largest eigenvalue. In the above case the
+ reciprocal condition number is about :math:`10^{-16}`. Which is
+ close to machine precision and even though the inverse exists, it
+ is meaningless, and care should be taken to interpet the results of
+ such an inversion.
+
+ Matrices with condition number lower than
+ ``min_reciprocal_condition_number`` are considered rank deficient
+ and by default Covariance::Compute will return false if it
+ encounters such a matrix.
+
+ a. ``use_dense_linear_algebra = false``
+
+ When performing large scale sparse covariance estimation,
+ computing the exact value of the reciprocal condition number is
+ not possible as it would require computing the eigenvalues of
+ :math:`J'J`.
+
+ In this case we use cholmod_rcond, which uses the ratio of the
+ smallest to the largest diagonal entries of the Cholesky
+ factorization as an approximation to the reciprocal condition
+ number.
+
+
+ However, care must be taken as this is a heuristic and can
+ sometimes be a very crude estimate. The default value of
+ ``min_reciprocal_condition_number`` has been set to a conservative
+ value, and sometimes the ``Covariance::Compute`` may return false
+ even if it is possible to estimate the covariance reliably. In
+ such cases, the user should exercise their judgement before
+ lowering the value of ``min_reciprocal_condition_number``.
+
+ b. ``use_dense_linear_algebra = true``
+
+ When using dense linear algebra, the user has more control in
+ dealing with singular and near singular covariance matrices.
+
+ As mentioned above, when the covariance matrix is near singular,
+ instead of computing the inverse of :math:`J'J`, the
+ Moore-Penrose pseudoinverse of :math:`J'J` should be computed.
+
+ If :math:`J'J` has the eigen decomposition :math:`(\lambda_i,
+ e_i)`, where :math:`lambda_i` is the :math:`i^\textrm{th}`
+ eigenvalue and :math:`e_i` is the corresponding eigenvector,
+ then the inverse of :math:`J'J` is
+
+ .. math:: (J'J)^{-1} = \sum_i \frac{1}{\lambda_i} e_i e_i'
+
+ and computing the pseudo inverse involves dropping terms from
+ this sum that correspond to small eigenvalues.
+
+ How terms are dropped is controlled by
+ `min_reciprocal_condition_number` and `null_space_rank`.
+
+ If `null_space_rank` is non-negative, then the smallest
+ `null_space_rank` eigenvalue/eigenvectors are dropped
+ irrespective of the magnitude of :math:`\lambda_i`. If the ratio
+ of the smallest non-zero eigenvalue to the largest eigenvalue in
+ the truncated matrix is still below
+ min_reciprocal_condition_number, then the
+ `Covariance::Compute()` will fail and return `false`.
+
+ Setting `null_space_rank = -1` drops all terms for which
+
+ .. math:: \frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}
+
+.. member:: int Covariance::Options::null_space_rank
+
+ Truncate the smallest ``null_space_rank`` eigenvectors when
+ computing the pseudo inverse of :math:`J'J`.
+
+ If ``null_space_rank = -1``, then all eigenvectors with eigenvalues
+ s.t.
+
+ :math:: \frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}
+
+ are dropped. See the documentation for
+ ``min_reciprocal_condition_number`` for more details.
+
+.. member:: bool Covariance::Options::apply_loss_function
+
+ Default: `true`
+
+ Even though the residual blocks in the problem may contain loss
+ functions, setting ``apply_loss_function`` to false will turn off
+ the application of the loss function to the output of the cost
+ function and in turn its effect on the covariance.
+
+.. class:: Covariance
+
+ :class:`Covariance::Options` as the name implies is used to control
+ the covariance estimation algorithm. Covariance estimation is a
+ complicated and numerically sensitive procedure. Please read the
+ entire documentation for :class:`Covariance::Options` before using
+ :class:`Covariance`.
+
+.. function:: bool Covariance::Compute(const vector<pair<const double*, const double*> >& covariance_blocks, Problem* problem)
+
+ Compute a part of the covariance matrix.
+
+ The vector ``covariance_blocks``, indexes into the covariance
+ matrix block-wise using pairs of parameter blocks. This allows the
+ covariance estimation algorithm to only compute and store these
+ blocks.
+
+ Since the covariance matrix is symmetric, if the user passes
+ ``<block1, block2>``, then ``GetCovarianceBlock`` can be called with
+ ``block1``, ``block2`` as well as ``block2``, ``block1``.
+
+ ``covariance_blocks`` cannot contain duplicates. Bad things will
+ happen if they do.
+
+ Note that the list of ``covariance_blocks`` is only used to
+ determine what parts of the covariance matrix are computed. The
+ full Jacobian is used to do the computation, i.e. they do not have
+ an impact on what part of the Jacobian is used for computation.
+
+ The return value indicates the success or failure of the covariance
+ computation. Please see the documentation for
+ :class:`Covariance::Options` for more on the conditions under which
+ this function returns ``false``.
+
+.. function:: bool GetCovarianceBlock(const double* parameter_block1, const double* parameter_block2, double* covariance_block) const
+
+ Return the block of the covariance matrix corresponding to
+ ``parameter_block1`` and ``parameter_block2``.
+
+ Compute must be called before the first call to ``GetCovarianceBlock``
+ and the pair ``<parameter_block1, parameter_block2>`` OR the pair
+ ``<parameter_block2, parameter_block1>`` must have been present in the
+ vector covariance_blocks when ``Compute`` was called. Otherwise
+ ``GetCovarianceBlock`` will return false.
+
+ ``covariance_block`` must point to a memory location that can store
+ a ``parameter_block1_size x parameter_block2_size`` matrix. The
+ returned covariance will be a row-major matrix.
+
+Example Usage
+-------------
+
+.. code-block:: c++
+
+ double x[3];
+ double y[2];
+
+ Problem problem;
+ problem.AddParameterBlock(x, 3);
+ problem.AddParameterBlock(y, 2);
+ <Build Problem>
+ <Solve Problem>
+
+ Covariance::Options options;
+ Covariance covariance(options);
+
+ vector<pair<const double*, const double*> > covariance_blocks;
+ covariance_blocks.push_back(make_pair(x, x));
+ covariance_blocks.push_back(make_pair(y, y));
+ covariance_blocks.push_back(make_pair(x, y));
+
+ CHECK(covariance.Compute(covariance_blocks, &problem));
+
+ double covariance_xx[3 * 3];
+ double covariance_yy[2 * 2];
+ double covariance_xy[3 * 2];
+ covariance.GetCovarianceBlock(x, x, covariance_xx)
+ covariance.GetCovarianceBlock(y, y, covariance_yy)
+ covariance.GetCovarianceBlock(x, y, covariance_xy)
+
